Diversification

BUSI 721, Fall 2022
JGSB, Rice University

Kerry Back

Why diversify your investments?

What is the risk of one coin flip for $10?
- The mean outcome is zero.
- The possible deviations from the mean are +10 and -10.

Variance is expected squared deviation from mean
- Flipping once for $10 $\Rightarrow$ variance = 100
- Flipping twice for $5 $\Rightarrow$ variance = (1/4) $\times$ 100 + (1/4) $\times$ 100 = 50
Standard deviation is square root of variance
- Flipping once for $10 $\Rightarrow$ std dev = 10
- Flipping twice for $5 $\Rightarrow$ std dev = 7.07

Suppose we start with $100 before we flip
There are two independent assets, each with possible returns of $\pm$ 10%
- Flipping once for $10 $\sim$ putting all $100 in a single asset
- Flipping twice for $5 $\sim$ putting $50 in each
Call the assets 1 and 2 with random returns $r_1$ and $r_2$
The portfolio return $r_p = (1/2) r_1 + (1/2) r_2$ is less risky than either $r_1$ or $r_2$.

Consider a $100,000 portfolio with 40% in one asset (Asset 1) and 60% in a second (Asset 2).
Suppose asset 1 $\uparrow$ 20%. $40,000 $\rightarrow$ $48,000
Suppose asset 2 $\uparrow$ 10%. $60,000 $\rightarrow$ $66,000
$100,000 $\rightarrow$ $114,000
So, up 14% = 0.4 $\times$ 20% + 0.6 $\times$ 10%
The portfolio return is $r_p = w_1r_1 + w_2r_2$ where $w_i$ is the fraction of the portfolio invested in asset $i$.

\[ \bar{r}_p = w_1 \bar{r}_1 + w_2 \bar{r}_2\]

\[w_1r_1 + w_2r_2 - (w_1 \bar{r}_1 +w_2\bar{r}_2)\]

\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]

$(a+b)^2 = a^2 + b^2 + 2ab$, so the squared deviation is the sum of $w_1^2(r_1-\bar{r}_1)^2$, $w_2^2(r_2 - \bar{r}_2)^2$, and $2w_1w_2(r_1-\bar{r}_1)(r_2-\bar{r}_2)$.

Expected value of $w_1^2(r_1-\bar{r}_1)^2$ is $w_1^2 \text{var}(r_1)$.
Expected value of $w_2^2(r_2 - \bar{r}_2)^2$ is $w_2^2 \text{var}(r_2)$.
Expected value of $2w_1w_2(r_1-\bar{r}_1)(r_2-\bar{r}_2)$ is $2w_1w_2\text{cov}(r_1,r_2)$.
Portfolio variance is

\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2) + 2 w_1 w_2 \text{cov}(r_1, r_2)\]

\[\left(\frac{1}{2}\right)^2 100\text{%}^2 + \left(\frac{1}{2}\right)^2 100\text{%}^2 = 50\text{%}^2\]

std dev of APPL is 0.01848158065635535

$var(r_{p})$ is smaller when the covariance is smaller.
Covariance is correlation times product of standard deviations, so $var(r_{p})$ is smaller when the covariance is smaller.

\[\small var(r_{p})=\sum_{i=1}^n w_{i}^2var(r_i)+2\sum_{i=1}^n \sum_{j=i+1}^n w_{i}w_{j}cov(r_{i},r_{j})\]

There are $\frac{n(n-1)}{2}$ covariance terms.

$n$ assets, all variances are the same = $\sigma^2$,
all correlations are the same = $\rho$,
all weights = $\frac{1}{n}$.

\[\small var(r_{p})=\sum_{i=1}^n (\frac{1}{n})^2\sigma^2+2\sum_{i=1}^n \sum_{j=i+1}^n (\frac{1}{n})^2\rho\sigma^2\]

\[\small var(r_{p})=\frac{1}{n}\sigma^2+\frac{n-1}{n}\rho\sigma^2 \rightarrow \rho\sigma^2\]

\[\small (w_{A}\sigma_{A}+w_{B}\sigma_{B})^2=w_{A}^2\sigma_{A}^2+w_{B}^2\sigma_{B}^2+2w_{A}w_{B}\sigma_{A}\sigma_{B}\]

same as $var_{p}$ except no correlation in the last term. So,

\[(w_{A}\sigma_{A}+w_{B}\sigma_{B})^2>var(r_{p})\]

Compare holding only asset B to holding some mix of A and B.

Substitute $w_{B}=(1-w_{A})$. Variance in general is

\[\small w_{A}^2\sigma_{A}^2+(1-w_{A})^2\sigma_{B}^2+2w_{A}(1-w_{A})\rho\sigma_{A}\sigma_{B}\]

Derivative with respect to $w_{A}$ evaluated at $w_{A}$=0:

\[\small 2w_{A}\sigma^2_{A}-2(1-w_{A})\sigma^2_{B}+2(1-2w_{A})\rho\sigma_{A}\sigma_{B}=-2\sigma^2_{B}+2\rho\sigma_{A}\sigma_{B}\]

Risk is lowered by $w_{A}$ $\uparrow$ when $\rho\sigma_{A}<\sigma_{B}$